Given:

• The universal set $U = \{ f, g, h, p, q, r \}$
• The set $B$ $= \{ g, q, r \}$

We need to find the following sets:

1. $B' \cup U$: This is the union of the complement of $B$ and $U$.

2. $B \cap \emptyset$: This is the intersection of $B$ with the empty set.

Solution:

1. $B' \cup U$:

The complement of set $B$ (denoted as $B'$) consists of all elements in $U$ that are not in $B$.

• $B = \{ g, q, r \}$
• The complement $B' = U - B = \{ f, h, p \}$

Now, the union of $B'$ and $U$ is:

$B' \cup U = \{ f, h, p \} \cup \{ f, g, h, p, q, r \} = \{ f, g, h, p, q, r \}$

So, $B' \cup U = U$, which is the entire universal set:

$B' \cup U = \{ f, g, h, p, q, r \}$

2. $B \cap \emptyset$:

The intersection of any set with the empty set is always the empty set. So:

$B \cap \emptyset = \emptyset$

Final Answers:

1. $B' \cup U = \{ f, g, h, p, q, r \}$
2. $B \cap \emptyset = \emptyset$

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Would you like further details or have any questions about the process?

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Here are some related questions for you:

1. What is the complement of a set, and how do we compute it?
2. Can you find the union of two sets that have no common elements?
3. How does the intersection of a set and the empty set always result in the empty set?
4. How do we represent sets in roster form and set-builder notation?
5. Can you calculate the complement of a set within a larger universal set if some elements are missing?

Tip: When finding complements, always be clear about the universal set you're working with, as the complement depends on that.